If $z_1 = 1+2i$ and $z_2 = 3+5i$ , then ${\mathop{\rm Re}\nolimits} \,\left( {\frac{{{{\overline Z }_2}{Z_1}}}{{{Z_2}}}} \right) = $

For any two complex numbers ${z_1}$and${z_2}$ and any real numbers $a$ and $b$; $|(a{z_1} - b{z_2}){|^2} + |(b{z_1} + a{z_2}){|^2} = $

$arg\left( {\frac{{3 + i}}{{2 - i}} + \frac{{3 - i}}{{2 + i}}} \right)$ is equal to

Find the complex number z satisfying the equations $\left| {\frac{{z - 12}}{{z - 8i}}} \right| = \frac{5}{3},\left| {\frac{{z - 4}}{{z - 8}}} \right| = 1$

If ${z_1},{z_2}$ and ${z_3},{z_4}$ are two pairs of conjugate complex numbers, then $arg\left( {\frac{{{z_1}}}{{{z_4}}}} \right) + arg\left( {\frac{{{z_2}}}{{{z_3}}}} \right)$ equals

Find the modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$